Abstract
We establish a new nonlinear retarded Volterra-Fredholm type integral inequality. The upper bounds of the embedded unknown functions are estimated explicitly by using the theory of inequality and analytic techniques. Moreover, an application of our result to the retarded Volterra-Fredholm integral equations for estimation is given.
1. Introduction
Gronwall-Bellman inequality [1, 2] is an important tool in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. A lot of its generalizations in various cases can be found from the literature (e.g., [3–7]). During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [8–18] and the references cited therein.
Gronwall-Bellman inequality [1, 2] can be stated as follows. If and are nonnegative continuous functions on an interval satisfying for some constant , then
In 2004, Pachpatte [9] has discussed the linear Volterra-Fredholm type integral inequality with retardation: In 2011, Abdeldaim and yakout [17] studied a new integral inequality of Gronwall-Bellman-Pachpatte type:
In this paper, on the basis of [9, 17], we discuss a new retarded nonlinear Volterra-Fredholm type integral inequality: where is a constant. The upper bound estimation of the unknown function is given by integral inequality technique, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation.
2. Main Result
Throughout this paper, denotes the set of real numbers, , , denotes the class of continuously differentiable functions defined on set with range in the set , denotes the class of continuous functions defined on set with range in the set , and denotes the derived function of a function .
We give the following notations used to simplify the details of presentation.
We technically define a sequence of functions by in (5), which can be defined recursively by Obviously, for all , the function is increasing and the sequence consists of nondecreasing nonnegative functions and satisfies , . Moreover, as defined in [4] for comparison of monotonicity of functions, because the ratios , , are all nondecreasing.
For given constant , we define functions which are strictly increasing. When there is no confusion, we simply let denote and denote its inverse.
We define functions :
We define function
Theorem 1. Suppose that is nondecreasing with and on ; all are continuous functions with for , , . Suppose that the function is increasing and has a solution for . If satisfies (5), then where are inverse functions of, respectively.
Proof. From (5) and (6), we have
for all . Let denote the function on the right-hand side of (13), which is a positive and nondecreasing function on . Then (13) is equivalent to
Differentiating with respect to , using (14), we have
by the monotonicity of and and the property of . From (16), we have
Integrating both sides of the above inequality from to , we obtain
for ; is chosen arbitrarily, where is defined by (8).
Let denote the function on the right-hand side of (18), which is a positive and nondecreasing function on . Then (18) is equivalent to
Differentiating with respect to , using (19), we haveby the monotonicity of and the property of . From (21), we have
for all . From (22), we havefor all , where is defined by (9). Repeating the same derivation as in (19), (23), and so on, we obtain
for all , where is defined by (9).
Let denote the function on the right-hand side of (24), which is a positive and nondecreasing function on . Then (24) is equivalent to
Differentiating with respect to , we have
for all . From (27), using (25), we have
for all , by the monotonicity of , and and the property of . Integrating both sides of the above inequality from to , we obtain
for all , where is defined by (9). Let denote the function on the right-hand side of (29), which is a positive and nondecreasing function on . Then (29) is equivalent to
Differentiating with respect to , using (30), we have
for all . From (32), we have
for all . Integrating both sides of the above inequality from to , we obtain
for all . From (19), (25), (30), and (34), we have
for all . Substituting (20), (26), and (31) into (35), we have
Since is chosen arbitrarily, we have
By the definition of and (15), we have
From (37) and (38), we have
or
By the definition of , the assumption of Theorem 1, and (40), we observe that
Since is increasing, from the last inequality and (14), we have the desired estimation (12).
We define the following functions:
for all , where , are defined by (8) and (9), respectively.
Corollary 2. Let , , , , , , be as in Theorem 1. Suppose that the function is increasing and has a solution for . If satisfies (5), then where are inverse functions of, respectively.
3. Application
In this section, we apply our result in Theorem 1 to investigate the retarded Volterra-Fredholm integral equations: for , where , is nondecreasing with ,,,,. Let ; then ,. Since , is an increasing and invertible function.
The following theorem gives the bound on the solution of (44).
Theorem 3. Suppose that , in (44) satisfy the conditions where ,,, and are as in Theorem 1; let . Assume that the function is increasing and has a solution for . If is a solution of (44), thenwhere ,,, and are as in Theorem 1.
Proof. Using the condition (45), we have for , where several changes of variables are made. Applying the result of Theorem 1 to the last inequality, we obtain the desired estimation (47).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The present investigation was supported, in part, by the National Natural Science Foundation of China (no. 11161018), in part, by the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and, in part, by the Natural Science Foundation of Fujian Province of China (no. 2012J01014). The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.